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Electron Beam Field
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Nominal Animal:
Note: I'm trying to provoke a correct intuitive picture here, and not be technically accurate.

When we think of a light ray, we think of something like a laser beam.  Such a beam can be collimated (so its cross section stays the same in perfect vacuum), or focused to a point.  We can generate beams with relatively large cross sections, with a desirable (usually Gaussian) distribution wrt. distance from the beam axis.  Larger cross section laser beams are used in holography, for example.

In many ways, electron beams behave in an analogous manner; and both exhibit particle-wave duality.  However, electrons are fermions and not bosons like photons are.  When the beam is very thin, or focused to a point, the electrons are easiest to control. 

It is when you have a wider diameter (larger cross section) high-intensity beam (relatively high density of electrons in the beam) that the differences start to really appear.  Unlike photons, electrons cannot occupy the same place and state at the same time, and trying to do so generates "pressure" that opposes that, diverging your beam. If you have a thin electron beam, that "pressure" is aligned on the axis of the beam, and can basically be ignored.  If you have the spatial/deflection control, you can use such a sharp beam to scan the target surface, with pretty good evenness.  If you want a wide electron beam, you essentially have an internal pressure you need to counter, while simultaneously controlling the collimation/focus of the beam, and the axial intensity distribution of the beam.

When doing lithography, you want the rays that passed through the mask to either be collimated (all have the same direction), or focused (so that the direction of each ray is a function of its location with respect to the target), so that the image created by the mask has sharp edges. (Perfectly collimated rays generate sharp edges from a flat mask to a flat surface, focused rays are sharpest from spherically curved masks to a spherically curved surface, at constant distances from the focal point.) Any variance in the ray directions is analogous to defocusing, and causes all edges to become "blurry".

(You can see this if you do shadow puppets.  Try using a 30cm × 30cm panel light, and all you see is fuzzy shadows.  Use a small pinpoint light, and you can get pretty sharp shadows.  You'd need a laser light panel -- collimated light from the entire panel surface -- to get a panel light to yield sharp shadows; but then, the shadow would be exactly the same size as the object, regardless of the distance.)

It reminds me of controlling high-pressure water jets.  With the correct nozzles, the jet looks more like a beam than anything else, until it hits something.  The hardware looks simple, but it really is more of an art.  Not because we cannot describe it mathematically, because we can; but because saying that you want something to behave thus and actually making it so, are two completely different things.  Proper engineering needed!
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